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\begin{verbatim}#|

 Copyright (C) 1994 by Yuan Yu.  All Rights Reserved.

 This script is hereby placed in the public domain, and therefore unlimited
 editing and redistribution is permitted.

 NO WARRANTY

 Yuan Yu PROVIDES ABSOLUTELY NO WARRANTY.  THE EVENT SCRIPT IS PROVIDED "AS IS"
 WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESS OR IMPLIED, INCLUDING, BUT NOT
 LIMITED TO, ANY IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
 PARTICULAR PURPOSE.  THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE
 SCRIPT IS WITH YOU.  SHOULD THE SCRIPT PROVE DEFECTIVE, YOU ASSUME THE COST OF
 ALL NECESSARY SERVICING, REPAIR OR CORRECTION.

 IN NO EVENT WILL Yuan Yu BE LIABLE TO YOU FOR ANY DAMAGES, ANY LOST PROFITS,
 LOST MONIES, OR OTHER SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT
 OF THE USE OR INABILITY TO USE THIS SCRIPT (INCLUDING BUT NOT LIMITED TO LOSS
 OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY THIRD
 PARTIES), EVEN IF YOU HAVE ADVISED US OF THE POSSIBILITY OF SUCH DAMAGES, OR
 FOR ANY CLAIM BY ANY OTHER PARTY.

|#

\end{verbatim}
\noindent{\sc Event}:   Start with the library {\tt{"mc20-2"}} using the compiled version.

\addvspace{18pt}
\begin{verbatim};          Proof of the Correctness of the MEMSET Function
#|
This is part of our effort to verify the Berkeley string library.  The 
Berkeley string library is widely used as part of the Berkeley Unix OS.

This is the source code of memset function in the Berkeley string library.

void *
memset(dst, c, n)
        void *dst;
        register int c;
        register size_t n;
{

        if (n != 0) {
                register char *d = dst;

                do
                        *d++ = c;
                while (--n != 0);
        }
        return (dst);
}

The MC68020 assembly code of the C function memset on SUN-3 is given as 
follows.  This binary is generated by "gcc -O".

0x29d0 <memset>:        linkw fp,#0
0x29d4 <memset+4>:      movel d2,sp@-
0x29d6 <memset+6>:      movel fp@(8),d2
0x29da <memset+10>:     movel fp@(16),d0
0x29de <memset+14>:     beq 0x29ec <memset+28>
0x29e0 <memset+16>:     moveal d2,a0
0x29e2 <memset+18>:     moveb fp@(15),d1
0x29e6 <memset+22>:     moveb d1,a0@+
0x29e8 <memset+24>:     subl #1,d0
0x29ea <memset+26>:     bne 0x29e6 <memset+22>
0x29ec <memset+28>:     movel d2,d0
0x29ee <memset+30>:     movel fp@(-4),d2
0x29f2 <memset+34>:     unlk fp
0x29f4 <memset+36>:     rts

The machine code of the above program is:

<memset>:     0x4e56  0x0000  0x2f02  0x242e  0x0008  0x202e  0x0010  0x670c
<memset+16>:  0x2042  0x122e  0x000f  0x10c1  0x5380  0x66fa  0x2002  0x242e
<memset+32>:  0xfffc  0x4e5e  0x4e75

'(78    86      0       0       47      2       36      46
  0     8       32      46      0       16      103     12
  32    66      18      46      0       15      16      193
  83    128     102     250     32      2       36      46
  255   252     78      94      78      117)
|#
; in the logic, the above program is defined by (memset-code).
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{memset-code} \\  
\=\+{\sc{memset-code}}\index{memset-code} \\ 
$=$$\;\;\;\;${\tt{'}}{\tt{(}}\=\+{\tt{78}}{\tt{ }}{\tt{86}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{47}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{36}}{\tt{ }}{\tt{46}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{46}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{16}}{\tt{ }}{\tt{103}}{\tt{ }}{\tt{12}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{66}}{\tt{ }}{\tt{18}} \\ 
{\tt{46}}{\tt{ }}{\tt{0}}{\tt{ }}{\tt{15}}{\tt{ }}{\tt{16}}{\tt{ }}{\tt{193}}{\tt{ }}{\tt{83}}{\tt{ }}{\tt{128}}{\tt{ }}{\tt{102}}{\tt{ }}{\tt{250}}{\tt{ }}{\tt{32}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{36}}{\tt{ }}{\tt{46}}{\tt{ }}{\tt{255}}{\tt{ }}{\tt{252}}{\tt{ }}{\tt{78}} \\ 
{\tt{94}}{\tt{ }}{\tt{78}}{\tt{ }}{\tt{117}}{\tt{)}}\-\-
\end{tabbing}

\begin{verbatim}; the computation time of the program.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{memset-t1} \\  
\=\+{\rm{memset-t1}}\,({\it{n\/}})\index{memset-t1} \\ 
$=$$\;\;\;\;$\=\+{\bf if }({\it{n\/}} $-\;1$) $=$ {\tt{0}}$\;\;${\bf then }{\tt{7}} \\ 
{\bf else }{\rm{splus}}\,({\tt{3}}, {\rm{memset-t1}}\,({\it{n\/}} $-\;1$)\index{memset-t1})\index{splus}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{memset-t} \\  
\=\+{\rm{memset-t}}\,({\it{n\/}})\index{memset-t} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $=$ {\tt{0}}$\;\;${\bf then }{\tt{9}} \\ 
{\bf else }{\rm{splus}}\,({\tt{7}}, {\rm{memset-t1}}\,({\it{n\/}})\index{memset-t1})\index{splus}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{verbatim}; an induction hint.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{memset-induct} \\  
\=\+{\rm{memset-induct}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memset-induct} \\ 
$=$$\;\;\;\;$\=\+{\bf if }({\it{n\/}} $-\;1$) $=$ {\tt{0}}$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{memset-induct}}\,(\=\+{\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn}, \\ 
{\rm{add}}\,({\tt{32}}, {\it{i*\/}}, {\tt{1}})\index{add}, \\ 
$1\;+$ {\it{i\/}}, \\ 
{\it{n\/}} $-\;1$, \\ 
{\rm{put-nth}}\,({\it{ch\/}}, {\it{i\/}}, {\it{lst\/}})\index{put-nth}, \\ 
{\it{ch\/}})\-\index{memset-induct}$\;${\bf  endif}\-\-
\end{tabbing}

\begin{verbatim}; the preconditions of the initial state.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{memset-statep} \\  
\=\+{\rm{memset-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memset-statep} \\ 
$=$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\it{s\/}})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;${\rm{evenp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{evenp} \\ 
$\wedge$$\;\;\;\;${\rm{rom-addrp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{38}})\index{rom-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mcode-addrp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\sc{memset-code}}\index{memset-code})\index{mcode-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{24}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\it{str\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\/}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\/}}, {\it{lst\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{24}}, {\it{str\/}}, {\it{n\/}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;$({\it{str\/}} $=$ {\rm{read-mem}}\,({\rm{add}}\,({\tt{32}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp}, {\tt{4}})\index{add}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{4}})\index{read-mem}) \\ 
$\wedge$$\;\;\;\;$({\it{ch\/}} $=$ {\rm{uread-mem}}\,({\rm{add}}\,({\tt{32}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp}, {\tt{11}})\index{add}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{1}})\index{uread-mem}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $=$ {\rm{uread-mem}}\,({\rm{add}}\,({\tt{32}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp}, {\tt{12}})\index{add}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{4}})\index{uread-mem}))\-\-
\end{tabbing}

\begin{verbatim}; an intermediate state.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{memset-s0p} \\  
\=\+{\rm{memset-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memset-s0p} \\ 
$=$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\it{s\/}})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;${\rm{evenp}}\,({\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{evenp} \\ 
$\wedge$$\;\;\;\;${\rm{rom-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{22}}, {\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{38}})\index{rom-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mcode-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{22}}, {\rm{mc-pc}}\,({\it{s\/}})\index{mc-pc})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\sc{memset-code}}\index{memset-code})\index{mcode-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an})\index{sub}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\tt{24}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{ram-addrp}}\,({\it{str\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\_\/}})\index{ram-addrp} \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{n\_\/}}, {\it{lst\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an})\index{sub}, {\tt{24}}, {\it{str\/}}, {\it{n\_\/}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;${\rm{equal*}}\,({\rm{read-an}}\,({\tt{32}}, {\tt{0}}, {\it{s\/}})\index{read-an}, {\rm{add}}\,({\tt{32}}, {\it{str\/}}, {\it{i*\/}})\index{add})\index{equal*} \\ 
$\wedge$$\;\;\;\;$({\it{str\/}} $=$ {\rm{read-dn}}\,({\tt{32}}, {\tt{2}}, {\it{s\/}})\index{read-dn}) \\ 
$\wedge$$\;\;\;\;$({\it{ch\/}} $=$ {\rm{nat-to-uint}}\,({\rm{read-dn}}\,({\tt{8}}, {\tt{1}}, {\it{s\/}})\index{read-dn})\index{nat-to-uint}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $=$ {\rm{nat-to-uint}}\,({\it{i*\/}})\index{nat-to-uint}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $=$ {\rm{nat-to-uint}}\,({\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\it{s\/}})\index{read-dn})\index{nat-to-uint}) \\ 
$\wedge$$\;\;\;\;$(({\it{i\/}} $+$ {\it{n\/}}) $\leq$ {\it{n\_\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$({\it{i*\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{n\_\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{nat-rangep}}\,({\it{i*\/}}, {\tt{32}})\index{nat-rangep} \\ 
$\wedge$$\;\;\;\;${\rm{uint-rangep}}\,({\it{n\_\/}}, {\tt{32}})\index{uint-rangep})\-\-
\end{tabbing}

\begin{verbatim}; from the intial state s to exit: s --> sn.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memset-s-sn \\ \index{memset-s-sn}
\=\+({\rm{memset-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memset-statep} $\wedge$ ({\it{n\/}} $=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;$({\rm{mc-pc}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{mc-pc} $=$ {\rm{rts-addr}}\,({\it{s\/}})\index{rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{read-dn} $=$ {\it{str\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{mc-mem}, {\it{n\/}}, {\it{lst\/}})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{15}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{add}}\,({\tt{32}}, {\rm{read-an}}\,({\tt{32}}, {\tt{7}}, {\it{s\/}})\index{read-an}, {\tt{4}})\index{add})\- \\ 
$\wedge$$\;\;\;\;$({\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{mc-rfile})\index{read-rn} $=$ {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memset-s-sn-rfile \\ \index{memset-s-sn-rfile}
\=\+(\=\+{\rm{memset-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memset-statep} \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$({\it{oplen\/}} $\leq$ {\tt{32}}) \\ 
$\wedge$$\;\;\;\;${\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memset-s-sn-mem \\ \index{memset-s-sn-mem}
\=\+(\=\+{\rm{memset-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memset-statep} \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{24}})\index{disjoint})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{9}})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} $=$ {\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-
\end{tabbing}

\begin{verbatim}; from the initial state s to s0: s --> s0.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memset-s-s0 \\ \index{memset-s-s0}
\=\+({\rm{memset-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memset-statep} $\wedge$ ({\it{n\/}} $\not=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;${\rm{memset-s0p}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn}, {\tt{0}}, {\tt{0}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\/}})\index{memset-s0p}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memset-s-s0-else \\ \index{memset-s-s0-else}
\=\+({\rm{memset-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memset-statep} $\wedge$ ({\it{n\/}} $\not=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{linked-rts-addr}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{linked-rts-addr} $=$ {\rm{rts-addr}}\,({\it{s\/}})\index{rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{linked-a6}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{linked-a6} $=$ {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{sub}}\,({\tt{32}}, {\tt{4}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub})\- \\ 
$\wedge$$\;\;\;\;$({\rm{rn-saved}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{rn-saved} $=$ {\rm{read-rn}}\,({\tt{32}}, {\tt{2}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memset-s-s0-rfile \\ \index{memset-s-s0-rfile}
\=\+({\rm{memset-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memset-statep} $\wedge$ ({\it{n\/}} $\not=$ {\tt{0}}) $\wedge$ {\rm{d3-7a2-5p}}\,({\it{rn\/}})\index{d3-7a2-5p}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memset-s-s0-mem \\ \index{memset-s-s0-mem}
\=\+(\=\+{\rm{memset-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memset-statep} \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{24}})\index{disjoint})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} $=$ {\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-
\end{tabbing}

\begin{verbatim}; from s0 to exit (base case), from s0 to s0 (induction case).
; base case: s0 --> exit.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memset-s0-sn-base \\ \index{memset-s0-sn-base}
\=\+({\rm{memset-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memset-s0p} $\wedge$ (({\it{n\/}} $-\;1$) $=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;$({\rm{mc-pc}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-pc} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{read-dn} $=$ {\it{str\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-mem}, {\it{n\_\/}}, {\rm{put-nth}}\,({\it{ch\/}}, {\it{i\/}}, {\it{lst\/}})\index{put-nth})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;$({\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-rfile})\index{read-rn} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{15}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{add}}\,({\tt{32}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}, {\tt{8}})\index{add})\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memset-s0-sn-rfile-base \\ \index{memset-s0-sn-rfile-base}
\=\+(\=\+{\rm{memset-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memset-s0p} \\ 
$\wedge$$\;\;\;\;$(({\it{n\/}} $-\;1$) $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;$({\it{oplen\/}} $\leq$ {\tt{32}}) \\ 
$\wedge$$\;\;\;\;${\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{d3-7a2-5p}}\,({\it{rn\/}})\index{d3-7a2-5p}$\;\;${\bf then }{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn} \\ 
{\bf else }{\rm{head}}\,({\rm{rn-saved}}\,({\it{s\/}})\index{rn-saved}, {\it{oplen\/}})\index{head}$\;${\bf  endif}\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memset-s0-sn-mem-base \\ \index{memset-s0-sn-mem-base}
\=\+(\=\+{\rm{memset-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memset-s0p} \\ 
$\wedge$$\;\;\;\;$(({\it{n\/}} $-\;1$) $=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\it{str\/}}, {\it{n\_\/}})\index{disjoint})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{7}})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} $=$ {\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-
\end{tabbing}

\begin{verbatim}; induction case: s0 --> s0.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memset-s0-s0 \\ \index{memset-s0-s0}
\=\+({\rm{memset-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memset-s0p} $\wedge$ (({\it{n\/}} $-\;1$) $\not=$ {\tt{0}})) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{memset-s0p}}\,(\=\+{\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn}, \\ 
{\rm{add}}\,({\tt{32}}, {\it{i*\/}}, {\tt{1}})\index{add}, \\ 
$1\;+$ {\it{i\/}}, \\ 
{\it{str\/}}, \\ 
{\it{n\/}} $-\;1$, \\ 
{\rm{put-nth}}\,({\it{ch\/}}, {\it{i\/}}, {\it{lst\/}})\index{put-nth}, \\ 
{\it{ch\/}}, \\ 
{\it{n\_\/}})\-\index{memset-s0p} \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\- \\ 
$\wedge$$\;\;\;\;$({\rm{linked-a6}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn})\index{linked-a6} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$({\rm{linked-rts-addr}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn})\index{linked-rts-addr} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{rn-saved}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn})\index{rn-saved} $=$ {\rm{rn-saved}}\,({\it{s\/}})\index{rn-saved}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memset-s0-s0-rfile \\ \index{memset-s0-s0-rfile}
\=\+({\rm{memset-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memset-s0p} $\wedge$ (({\it{n\/}} $-\;1$) $\not=$ {\tt{0}}) $\wedge$ {\rm{d3-7a2-5p}}\,({\it{rn\/}})\index{d3-7a2-5p}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memset-s0-s0-mem \\ \index{memset-s0-s0-mem}
\=\+(\=\+{\rm{memset-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memset-s0p} \\ 
$\wedge$$\;\;\;\;$(({\it{n\/}} $-\;1$) $\not=$ {\tt{0}}) \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\it{str\/}}, {\it{n\_\/}})\index{disjoint})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\tt{3}})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} $=$ {\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-
\end{tabbing}

\begin{verbatim}; put together (s0 --> exit).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memset-s0-sn \\ \index{memset-s0-sn}
\=\+{\bf let }\=\+{\it{sn\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{memset-t1}}\,({\it{n\/}})\index{memset-t1})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+{\rm{memset-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memset-s0p} \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\it{sn\/}})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;$({\rm{mc-pc}}\,({\it{sn\/}})\index{mc-pc} $=$ {\rm{linked-rts-addr}}\,({\it{s\/}})\index{linked-rts-addr}) \\ 
$\wedge$$\;\;\;\;$({\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\it{sn\/}})\index{read-dn} $=$ {\it{str\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str\/}}, {\rm{mc-mem}}\,({\it{sn\/}})\index{mc-mem}, {\it{n\_\/}}, {\rm{memset1}}\,({\it{i\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memset1})\index{mem-lst} \\ 
$\wedge$$\;\;\;\;$({\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{sn\/}})\index{mc-rfile})\index{read-rn} $=$ {\rm{linked-a6}}\,({\it{s\/}})\index{linked-a6}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{15}}, {\rm{mc-rfile}}\,({\it{sn\/}})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{add}}\,({\tt{32}}, {\rm{read-an}}\,({\tt{32}}, {\tt{6}}, {\it{s\/}})\index{read-an}, {\tt{8}})\index{add})\-)\-\-$\;${\bf  endlet}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memset-s0-sn-rfile \\ \index{memset-s0-sn-rfile}
\=\+({\rm{memset-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memset-s0p} $\wedge$ ({\it{oplen\/}} $\leq$ {\tt{32}}) $\wedge$ {\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\rm{stepn}}\,({\it{s\/}}, {\rm{memset-t1}}\,({\it{n\/}})\index{memset-t1})\index{stepn})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{d3-7a2-5p}}\,({\it{rn\/}})\index{d3-7a2-5p}$\;\;${\bf then }{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn} \\ 
{\bf else }{\rm{head}}\,({\rm{rn-saved}}\,({\it{s\/}})\index{rn-saved}, {\it{oplen\/}})\index{head}$\;${\bf  endif}\-)\-\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  memset-s0-sn-mem \\ \index{memset-s0-sn-mem}
\=\+({\rm{memset-s0p}}\,({\it{s\/}}, {\it{i*\/}}, {\it{i\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}}, {\it{n\_\/}})\index{memset-s0p} $\wedge$ {\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\it{str\/}}, {\it{n\_\/}})\index{disjoint}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\rm{stepn}}\,({\it{s\/}}, {\rm{memset-t1}}\,({\it{n\/}})\index{memset-t1})\index{stepn})\index{mc-mem}, {\it{k\/}})\index{read-mem} \\ 
$=$$\;\;\;\;${\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-\-
\end{tabbing}

\begin{verbatim}; the correctness of the MEMSET program.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  memset-correctness \\ \index{memset-correctness}
\=\+{\bf let }\=\+{\it{sn\/}}{\bf $\;$ be$\;\;$}{\rm{stepn}}\,({\it{s\/}}, {\rm{memset-t}}\,({\it{n\/}})\index{memset-t})\index{stepn}\- \\ 
{\bf in} \\ 
\=\+{\rm{memset-statep}}\,({\it{s\/}}, {\it{str\/}}, {\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memset-statep} \\ 
$\rightarrow$$\;\;\;\;$(\=\+({\rm{mc-status}}\,({\it{sn\/}})\index{mc-status} $=$ {\tt{'}}{\tt{running}}) \\ 
$\wedge$$\;\;\;\;$({\rm{mc-pc}}\,({\it{sn\/}})\index{mc-pc} $=$ {\rm{rts-addr}}\,({\it{s\/}})\index{rts-addr}) \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{sn\/}})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\tt{32}}, {\tt{14}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\- \\ 
$\wedge$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\tt{32}}, {\tt{15}}, {\rm{mc-rfile}}\,({\it{sn\/}})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{add}}\,({\tt{32}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp}, {\tt{4}})\index{add})\- \\ 
$\wedge$$\;\;\;\;$(\=\+(({\it{oplen\/}} $\leq$ {\tt{32}}) $\wedge$ {\rm{d2-7a2-5p}}\,({\it{rn\/}})\index{d2-7a2-5p}) \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{sn\/}})\index{mc-rfile})\index{read-rn} \\ 
$=$$\;\;\;\;${\rm{read-rn}}\,({\it{oplen\/}}, {\it{rn\/}}, {\rm{mc-rfile}}\,({\it{s\/}})\index{mc-rfile})\index{read-rn})\-)\- \\ 
$\wedge$$\;\;\;\;$(\=\+(\=\+{\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\rm{sub}}\,({\tt{32}}, {\tt{8}}, {\rm{read-sp}}\,({\it{s\/}})\index{read-sp})\index{sub}, {\tt{24}})\index{disjoint} \\ 
$\wedge$$\;\;\;\;${\rm{disjoint}}\,({\it{x\/}}, {\it{k\/}}, {\it{str\/}}, {\it{n\/}})\index{disjoint})\- \\ 
$\rightarrow$$\;\;\;\;$(\=\+{\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{sn\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem} \\ 
$=$$\;\;\;\;${\rm{read-mem}}\,({\it{x\/}}, {\rm{mc-mem}}\,({\it{s\/}})\index{mc-mem}, {\it{k\/}})\index{read-mem})\-)\- \\ 
$\wedge$$\;\;\;\;$({\rm{read-dn}}\,({\tt{32}}, {\tt{0}}, {\it{sn\/}})\index{read-dn} $=$ {\it{str\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{mem-lst}}\,({\tt{1}}, {\it{str\/}}, {\rm{mc-mem}}\,({\it{sn\/}})\index{mc-mem}, {\it{n\/}}, {\rm{memset}}\,({\it{n\/}}, {\it{lst\/}}, {\it{ch\/}})\index{memset})\index{mem-lst})\-\-$\;${\bf  endlet}\-
\end{tabbing}


\noindent{\sc Event}:   Disable memset-t.

\addvspace{18pt}

\begin{verbatim}; some properties of memset.
; see file cstring.events.
\end{verbatim}
%\end{list}
\printindex
\end{document}
